From: asimov@nas.nasa.gov (Daniel A. Asimov)
Newsgroups: sci.math.research
Subject: Re: Realizability in R^3 of Simplicial Orientable Surfaces
Date: Tue, 20 Jun 1995 23:28:43 GMT

In article <DAHoMr.3B1@cnn.nas.nasa.gov> asimov@nas.nasa.gov (Daniel A. Asimov) writes:
>In the book "Convex Polytopes" by Branko Grunbaum (1967), the following
>conjecture is described as neither proved nor disproved:
>
>  Given any finite simplicial complex K whose underlying topological space is a 
>  compact orientable surface without boundary, then K can be realized as an 
>  embedded piecewise linear subset of R^3.  
>
>(An easy argument shows that any such K, even if non-orientable, can be so 
>realized in R^5.)
>
>
>QUESTIONS:	
>
>1.	Is this conjecture still unresolved?
>
>2.	What about the non-orientable case?  Is it possible that all such 
>	non-orientable K can be so realized in R^4 ?
>
>Daniel Asimov
>
>Mail Stop T27A-1
>NASA Ames Research Center
>Moffett Field, CA 94035-1000
>
>asimov@nas.nasa.gov
>(415) 604-4799 w
>(415) 604-3957 fax
>
>[I assume that Grunbaum means an embedding which is linear on each
>simplex; if the embedding is merely PL on each simplex then the 
>question is trivial.  -Greg]

P.S.  Yes, indeed, I mean that each simplex of K is embedded affinely in R^3.
Grunbaum is not to blame; the above is my own paraphrase of his statement.
--Dan